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Banach–Tarski Paradox
- maze-le 2y agoThis must be the most unintuitive result of all of mathematics. Its very interesting what a seemingly simple axiom like the axiom of choice can lead to -- simple as in 'even a 9-year old can understand it', the consequences are rather enormous and not simple at all.
- tsimionescu 2y agoHonestly, it's not that surprising if you learned the properties of infinity before, especially of uncountable infinity. If 2*Inf == Inf, and if a sphere has an infinity of points, it's not that surprising that you can make two spheres from those same points. The construction itself is of course much more impressive, I'm not downplaying it, but I don't think it's less intuitive than other properties of infinity. My personal reckoning with this was learning that there are as many numbers in the [0,1] interval of the real line as on the whole real line.
- bubblyworld 2y agoThe BT paradox includes the requirement that the pieces are separated and put back together using isometries of R3, which is _way_ more restrictive than isomorphism of sets (what you're talking about). So it's quite surprising from that point of view!
- petters 2y agoReally? I think this is on a completely different level of intuition. There are five pieces here that are only rotated and translated.
- stared 2y ago"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" Well, the axiom of choice gives a lot of counterintuitive examples, with the Banach-Tarski paradox being the easiest to imagine by a non-mathematician. Yet, I know no consequences that would be measurable in physics. To my knowledge, AoC is more like glue, which (paradoxically) makes quite a few things smoother, e.g., all Hilbert spaces have a basis. Otherwise one runs in a lot of theorems, in all corners of maths, with "this is always true for finite, for infinite we know that there are no counterexamples, yet we cannot prove that for all cases".
- dist-epoch 2y agoMy understanding is the general consensus is that no physical infinity can exist. So Axiom of Choice/Banach-Tarski doesn't really apply in physics since they are only interesting when talking about infinite sets.
- nairboon 2y agoIn what sense could it exsist then, if infinity is not physically realizable? Does infinity even exist?
- dist-epoch 2y ago> David Hilbert famously argued that infinity cannot exist in physical reality. The consequence of this statement — still under debate today — has far-reaching implications. https://www.nature.com/articles/s41567-018-0238-1 https://www.nature.com/articles/s41567-018-0238-1
- stared 2y agoWell, in principle, the Universe can be infinite. Sure, we cannot measure infinity, but to be fair, all mathematical concepts (when looked at closely enough) are not something we measure directly. Even if a kindergarten-level maths of "there are three apples," we do an abstraction. We need to decide that something is a separate object, an apple (how big or small should a fruit be an apple? if there is a bite, is it an apple? etc, etc) - usually with an assumption that all apples are the same (which we know is not true, but serves as an useful approximation). pretend that
- thechao 2y agoAs far as we can tell, GR implies, and we have measured, space-time is completely continuous. Draw a square on a piece of paper; or, better yet, outline a cube with some sticks: within that square (or cube) is an infinite set of points of either the integral or real cardinality — whichever you’d like. The “no physical infinity” thing sounds like a very Greek sort of axiom — like their “nature abhors a vacuum” thing, etc.
- carlos-menezes 2y agoMandatory watch, by Vsauce: https://www.youtube.com/watch?v=s86-Z-CbaHA https://www.youtube.com/watch?v=s86-Z-CbaHA
- smusamashah 2y agoAn aside, I have never seen anyone write the digit 8 like that before. My takeaway from the video is that real numbers between 0-1 or 1-2 are infinite. And infinity+1 or -1 is still infinity. Even if you take something out from between 1-2, it will remain the same.
- gus_massa 2y agoI was going to recomend that video. It has a good description of the technical details, but it uses a graphical representation to make it clear. But it's not a stupid graphical representation that hides all the details under flashy animations.
- adastra22 2y agoCan someone explain this more simply? If you cut up the sphere's surface into pieces, the combined surface area will remain the same. If you then reassemble them in a different configuration into two spheres both the same size as the original, the surface area will be twice as much. I don't see how that could be true. What am I missing here? ETA: thanks for all the explanations. The most succinct answer seems to be because it assumes the surface is made of infinitely many points, and infinity breaks math. 2*inf = inf. One more reason why it makes no sense to treat infinity as a number.
- pangratz 2y agoVSauce did a nice video on this, starting at 15m25s with the sphere re-assembling: https://www.youtube.com/watch?v=s86-Z-CbaHA&t=15m25s https://www.youtube.com/watch?v=s86-Z-CbaHA&t=15m25s
- djvdq 2y agoI don't know if this will explain your questions, but years ago I watched this video from Vsauce about this very topic, and IIRC it was explained quite nicely: https://www.youtube.com/watch?v=s86-Z-CbaHA https://www.youtube.com/watch?v=s86-Z-CbaHA
- bheadmaster 2y agoThis sentence seems crucial: It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices. So you chop up a sphere (which has a volume V1) into a finite number of collections of points (which don't have a volume), then assemble the collections of points into a new sphere (which has a volume V2 != V1).
- tsimionescu 2y agoYou chop up a sphere of volume V1 into non-measurable pieces, and then re-assemble those pieces into 2 spheres, each of volume V1.
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- ragtagtag 2y agoWhat's an anagram of Banach-Tarski? Banach-Tarski Banach-Tarski!
- throwaway81523 2y agoIf we're doing dad jokes now, it's unfortunate that King Solomon didn't know about the Banach-Tarski paradox. Otherwise, instead of cutting the baby into two pieces, he could have suggested five pieces and made both mothers happy.
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- RandomLensman 2y agoIf I remember the paper correctly, it also uses a metric that isn't just the usual euclidean one.
- ykonstant 2y agoThe gorgeous book "Discrete groups, expanding graphs and invariant measures" by A. Lubotzky investigates the structures that give rise to measure-theoretic phenomena like the B-T paradox. It is a graduate level monograph, but I recommend it wholeheartedly. It illustrates how the study of some paradoxes from the early 20th Century led to amazing and highly applicable mathematics like expander graphs and the spectral theory of non-commutative groups.
- aquafox 2y agoThere's a short proof of a 2D version of the paradox on page 684 of the Princeton Companion to Mathematics: https://sites.math.rutgers.edu/~zeilberg/akherim/PCM.pdf https://sites.math.rutgers.edu/~zeilberg/akherim/PCM.pdf
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- ColinWright 2y agoThis is a fabulous result, both positive and negative, for many reasons. But one of the things people don't realise is that there is a reason why it's interesting mathematically and not just a gimmick. In Classical Euclidean Geometry there are five axioms, and while the first four seem clear and obvious, the fifth seems a little contrived. So for centuries people tried to prove that the fifth was unnecessary and could be proven from the other four. These attempts all failed, and we can show that they must fail, because there are systems that satisfy the first four, but do not satisfy the fifth. Hence the fifth cannot be a consequence of the first four. Such systems are (for obvious reasons) called Non-Euclidean Geometries. So we can use explicit examples to demonstrate that certain proofs are impossible, and the Banach-Tarski Theorem is a result that proves that a "Measure"[0] cannot have all four obviously desirable characteristics. For more information, here's a blog post[1] I wrote some time ago: https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheorem.html?xf26hn https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheo... It's intended to be readable, but the topic is inherently complex, so it may need more than one read through. If you're interested. [0] Technical term for a function that takes an object and returns a concept of its size. For lines it's length, for planar objects it's area, for 3D objects it's volume, and so on. [1] In case people want to discuss that separately I've submitted it as a separate post here: https://news.ycombinator.com/item?id=40798224 https://news.ycombinator.com/item?id=40798224
- lisper 2y agoNice write-up, though I wish you had expanded a bit more on this: "We can show that using the axioms of Zermelo-Fraenkel Set Theory we cannot prove the product of an infinite collection of non-empty sets to be non-empty. That seems daft..." Indeed, and it's exactly the sort of thing that you should not simply proclaim to be true with no explanation or reference. BTW, you might find this interesting: https://blog.rongarret.info/2023/01/an-intuitive-counterexample-to-axiom-of.html https://blog.rongarret.info/2023/01/an-intuitive-counterexam...
- ColinWright 2y ago> Nice write-up, Thank you. > ... I wish you had expanded a bit more on this: >> "We can show that using the axioms of Zermelo-Fraenkel Set Theory we cannot prove the product of an infinite collection of non-empty sets to be non-empty. That seems daft..." > Indeed, and it's exactly the sort of thing that you should not simply proclaim to be true with no explanation or reference. I'll see if I can hunt out an accessible reference. Problem is, I think it uses Cohen's forcing, and that's pretty tough going in any kind of detail. I might add a note ... thanks for the suggestion. > BTW, you might find this interesting: https://blog.rongarret.info/2023/01/an-intuitive-counterexample-to-axiom-of.html https://blog.rongarret.info/2023/01/an-intuitive-counterexam... I've bookmarked that for when I have a coffee, biscuit, and 30 minutes. Thank you.
- IngoBlechschmid 2y agoThere are three ways to resolve this paradox: 1. Accept that our intuition about volumes is off when dealing with point clouds so weird that they cannot actually be described, but require the axiom of choice to concoct them. 2. Reject the axiom of choice and adopt the axiom of determinacy. This axiom restores our intuition about volumes to all subsets of Euclidean space, at the expense of which sets can be formed. (That said, the axiom of determinacy allows other sets to be formed which are not possible with the axiom of choice, so it wouldn't be correct to state that the axiom of determinacy causes the set-theoretic universe to shrink.) 3. Keep logic and set theory as it is, but employ locales instead of topological or metric spaces. Locales are an alternative formalization of the intuitive notion of spaces. For many purposes, there are little differences between locales and more traditional sorts of spaces. But, crucially, a locale can be nontrivial even if it does not contain any points. Locale-theoretically, the five pieces appearing in the Banach–Tarski paradox have a nontrivial overlap (even though no points are contained in the overlapping regions), hence you wouldn't expect the volumes to add up. I tried to give a varied account on the axiom of choice at the Chaos Communication Congress once, the slides are here: https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choice.pdf https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choic...
- constantcrying 2y agoWhy didn't you list the actual "standard" approach of accepting that certain sets have no definable volume? This is the basis of measure theory, which is perfectly accepting of the fact that measures don't need to be defined everywhere.
- IngoBlechschmid 2y agoSorry, I was in a hurry before, the standard approach is exactly what I wanted to refer to with option 1!
- aaron695 2y agoA. K. Dewdney did a Computer Recreations on this - "A matter fabricator provides matter for thought" on the hub - DOI:10.2307/24987222 ( https://www.jstor.org/stable/24987222 https://www.jstor.org/stable/24987222 ) [Early April 1989] It made quite an impression as a kid. Even 30 years later I think about it every now and again.
- hackandthink 2y ago"Tame topology is the name for the largely programmatic quest for a refoundation of topology and geometry that avoids ‘pathological’ objects like space-filling curves or counter-intuitive results like the Banach-Tarski paradox that occur in the traditional approach." https://ncatlab.org/nlab/show/tame+topology https://ncatlab.org/nlab/show/tame+topology
- senorqa 2y agoWhat's the use of this paradox? Does it have any practical implementation?
- constantcrying 2y agoIt gave rise to measure theory, which is now an extremely important mathematical theory and the basis for analysis and stochastics.
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